Best Known (35, 35+9, s)-Nets in Base 81
(35, 35+9, 2362874)-Net over F81 — Constructive and digital
Digital (35, 44, 2362874)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (7, 11, 265724)-net over F81, using
- net defined by OOA [i] based on linear OOA(8111, 265724, F81, 4, 4) (dual of [(265724, 4), 1062885, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(8111, 531448, F81, 4) (dual of [531448, 531437, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(814, 531441, F81, 2) (dual of [531441, 531437, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(8111, 531448, F81, 4) (dual of [531448, 531437, 5]-code), using
- net defined by OOA [i] based on linear OOA(8111, 265724, F81, 4, 4) (dual of [(265724, 4), 1062885, 5]-NRT-code), using
- digital (24, 33, 2097150)-net over F81, using
- net defined by OOA [i] based on linear OOA(8133, 2097150, F81, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8133, 8388601, F81, 9) (dual of [8388601, 8388568, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8133, 8388601, F81, 9) (dual of [8388601, 8388568, 10]-code), using
- net defined by OOA [i] based on linear OOA(8133, 2097150, F81, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- digital (7, 11, 265724)-net over F81, using
(35, 35+9, large)-Net over F81 — Digital
Digital (35, 44, large)-net over F81, using
- t-expansion [i] based on digital (33, 44, large)-net over F81, using
- 1 times m-reduction [i] based on digital (33, 45, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- 1 times m-reduction [i] based on digital (33, 45, large)-net over F81, using
(35, 35+9, large)-Net in Base 81 — Upper bound on s
There is no (35, 44, large)-net in base 81, because
- 7 times m-reduction [i] would yield (35, 37, large)-net in base 81, but